# Bayesian inference with a binary and a continuous variable

I am studying the link between a linguistic feature (the specifics doesn't really matter but for any given language there are two options, the language either has the feature or it doesn't) and gender disparity in a given country. I am planning to only include countries where a significant portion of the population (say at least 75%) speaks a single language to make things easier.

Gender disparity is extremely difficult to quantify but I have looked at the Gender Inequality Index published by the UN (though it severely punishes economically underdeveloped countries on the basis of access to hygiene products), percentage of representatives who do not identify as a woman in the countries' national legislatures (it could be argued that this is not a metric of gender disparity but a lower percentage score here is usually an indicator better equality and I would not dream of using this data on its own), the gender gap in literacy rates (though this only matters in countries where there is illiteracy to begin with), and finally data from World Values Survey where I will take the percentage of the respondents who agreed (strongly or not) with a statement that was discriminatory against women for a given country.

Now the problem is, I want to specifically argue that there isn't a positive correlation between the linguistic feature and gender disparity. My supervisor told me that I would have to use Bayesian inference as not finding evidence for the alternative hypothesis does not necessarily mean that my H0 is true. I did some reading on Bayesian inference, namely Bayesian Statistics for Beginners: a step-by-step approach by Therese Donovan since stats is neither my nor my supervisor's forte, and from what I understand given the sample data that I have attached to this thread I calculated the posterior probabilities of the two hypotheses (H1: a positive correlation between the two variables, H2: no positive correlation between the two variables), My definition of likelihood is problematic, I believe, because I simply calculated the likelihood of any new country that I add to the dataset speaking a language that has the linguistic feature in question and has gender inequality values above the average/median inequality values of the country where a language that doesn't have the feature is spoken for the likelihood of H1. For example: In the sample dataset above I have a total of 10 countries, the probability of a new country speaking a language with an EGM value of 1 and a percentage score higher than the median (I used median here because using averages gave a 0% chance for the probability of H1, but I am unsure whether to use mean of median in the larger dataset) of languages with an EGM value of 0 is 0.1 as out of the 10 countries only 1 fits this profile. Whereas the likelihood of H2 is calculated as all other scenarios, the new language added having a lower or equal score than the average/median of EGM=0 languages, which in our sample is 0.4. I then calculated the posterior probabilities using the following equation: Based on the values at hand I feel like I can say that there is an 80% chance that there is no positive correlation between the linguistic feature and gender inequality. My supervisor found this viable and advised that I get the statistical setup checked by the statistician in our department. But I wanted to come here and get a better understanding before I go ahead and do that.

My questions are:

1. My data are most likely not normally distributed, would that cause any issues in my analysis?

2. I can see that there is, surprisingly, a negative correlation between the feature and gender inequality from the box plots I created using the sample, which works to my advantage as I only argue that there isn't a positive one. Is there any way I could quantify the extent of the correlation? I saw a couple of posts here that advised using point-biserial correlation when one of the variables is binary, as is the case here.

3. Is my Bayesian inference setup right? In particular, am I right in thinking that the likelihood for H1 (positive corr.) is the likelihood of finding countries that are both EGM=1 and inequality values greater than the average of countries that are EGM=0, and that the likelihood for H0 (no positive corr.) is the likelihood of finding countries that are both EGM=1 and inequality values equal or below the average of countries that are EGM=0. The reason for this definition is that the more countries I find that are EGM=1 and have values greater than the average of EGM=0, there will come a point where EGM=1 average will be greater than EGM=0 average which would mean that there is a positive correlation between the feature and inequality since correlation between a binary variable and a continuous variable is a comparison of averages.

4. This is linked to the previous question but is there any reason I am using averages for the comparison and not medians? Using median values to get rid of outliers seems like a good idea in theory but there must be a reason that makes averages the superior choice in a comparison like this.

This is my first time posting here so forgive me if my post is extremely long and/or not suitable for the platform as the only reason I am doing this is to get some prior background on the subject before I face the department statistician so that I will not look like a complete fool with no understanding of even the most basic statistical concepts. If it matters, I only took what they called 'Advanced Statistics' as part of my Economics degree more than 3 years ago, and have no prior or posterior exposure to statistics apart from interpreting the results of other academic studies. It just seems there there are an endless number of statistical tools and I am afraid of using a hammer to screw together two pieces of wood. Anyway, many thanks in advance for all of your responses!